13.3 Summary [particle electrophoresis top]

In this chapter, we have highlighted that particle electrophoresis comprises the same physics as electroosmosis, with the key differences being (1) that the coordinate system is typically one in which the fluid is motionless, rather than the particle, (2) the curvature of the interface cannot be ignored if double layers are not thin, and (3) the electrical double layer exhibits two-way coupling with the fluid flow if the surface potential is large. Particle electrophoresis transitions analytically between our early study of equilibrium electrokinetics with one-way coupling (electroosmosis with thin double layers) and our later study of electrokinetics with dynamic electric fields (e.g., induced-charge phenomena; Chapter 16) or with two-way coupling (e.g., nanofluidic devices with varying channel cross-sections; Chapter 15).

For small surface potentials and thin double layers, the electrophoretic mobility of a particle of any shape is the same as the electroosmotic mobility of a wall made of the same material—at steady-state, the two processes are identical except for a coordinate transformation:

For particles with small surface potentials but finite double layers, the extension of the double layer causes a reduction in electrophoretic velocity since the net charge density in the double layer experiences lower local electric fields than in the thin double layer case. For spheres with small surface potentials but finite double layers, we found that Henry’s function:

describes the correction factor, which reduces the electrophoretic velocity to two thirds of the thin double layer result as the double layers become thick. For particles with large surface potentials, the system exhibits two-way coupling, and the flow perturbs the electrical double layer. When this happens, the motion of counterions and coions suppresses the local electric field, further reducing the electrophoretic mobility. For spheres, this has been numerically calculated in detail and approximate relations were presented. For more complicated shapes, the electrophoretic mobility can be calculated only with a full analysis of the ion transport in the system using the Poisson, Nernst-Planck, and Navier-Stokes equations.

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